Let’s take a class vote. How many of you are registered to vote?

How groups can best arrive at decisions (goals). find an out come that “reflects the will of the people.” What do you think is the best way for millions of people to make one decision?

Preference List Ballot (def) A ballot consisting of a rank ordering of candidates. Usually in the form of a vertical list with our most preferred candidate on top. No ties allowed.

Examples of Preference List Ballot Australia uses it too!

The Number of Voters Assumption (rule) For the sake of avoiding excessive annoyances in the theory… Throughout the chapter, we will assume that the number of voters is odd. In reality, numbers of voters are often so high that ties are unlikely anyways.

Majority Rule The most obvious of voting methods… 3 desirable properties:

Do we take these simple traits for granted?? The first desirability fails in the system called a Dictatorship, where all ballots except the dictators are ignored… The second desirability fails in imposed rule, where a certain candidate wins regardless of who votes for whom … The third desirability fails in minority rule, where fewest votes win…

May’s Theorem (Kenneth May, 1952)

Who would beat whom if two candidates faced an election 2 at a time? RankNumber of Voters (3) firstABC secondBCA thirdCAB A would defeat B (2:1) B would defeat C (2:1)

Condorcet’s Method (procedure) Extending majority rule to three or more candidates…

Example- Condorcet’s Method Rank6531 firstGBAGRNPB secondAGRNAGGB thirdPBGB AG fourthRNPB RN Number of voters (15) Amount of voters that voted in the same order. Who is the winner using Condorcet’s Method? AG too good to be true. There is one tragic flaw…

Condorcet’s Voting Paradox

Remember this slide??? RankNumber of Voters (3) firstABC secondBCA thirdCAB A would defeat B (2:1) B would defeat C (2:1) C would defeat A (2:1)

Explain why it’s impossible to have two winners using Condorcet’s method with an odd amount of voters By definition of the method, a person wins by beating all others in head to head elections. Since amount of voters is odd, no one head-to-head election will have a tie. How can B beat A if A has already been determined to beat all other candidates??

Plurality Voting Can you see a potential problem with this method???

1980 Senate Race in NY 22%23%15%29%7%4% DDHHJJ HJDJHD JHJDDH D- Alfonse D’Amato- Conservative H- Elizabeth Holtzman- Liberal J- Jacob Javitz- Liberal Is there a Condorcet winner? Who won using Plurality Voting? Yes, Elizabeth Holtzman Yes, Alfonse D’Amato

Condorcet Winner Criterion

Manipulability Can you think of an example of this? It is a problem for Plurality voting, but not the Condorcet Method

Which is your favorite to win the tournament?

Rank Methods (procedure) Condorcet’s contemporary, Jean-Charles de Borda ( )

Borda Count (procedure) How much is the last place vote worth??

Borda Count Failure Independence of Irrelevant Alternatives (IIA) if it is impossible for a candidate B to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had B and the winning candidate ranked.

Independence of Irrelevant Alternatives

Show that the Borda Count Method does not satisfy IIA. RankNumber of Voters (5) firstAAACC secondBBBBB thirdCCCAA RankNumber of Voters (5) firstAAABB secondBBBCC thirdCCCAA

Sequential Pairwise Voting (procedure) Start with an agenda and pair the first two candidates in a one-on-one contest. The winner moves on to confront the third candidate in the list, one-on-one. Losers are deleted. The candidate remaining at the end wins.

Example: Sequential Pairwise Voting RankNUMBER OF VOTERS (3) firstACB secondBAD thirdDBC fourthCDA Who Wins? Anything wrong with that?

Pareto Condition (failed by SPV method)

The Hare System (procedure) Repeatedly delete candidates that are “least preferred” in the sense of being at the top of the fewest ballots. Number of Voters (13) Rank5431 firstACBB secondBBCA thirdCAAC B and C are both eliminated in the first round

Hare System Number of Voters (13) Rank5431 firstACBA secondBBCB thirdCAAC Suppose the voter in the last column moved A up in his or her vote. The only change made is favorable to A.

Hare System Number of Voters (13) Rank5431 firstACBA secondBBCB thirdCAAC Reapply the Hare system and see that only B is eliminated in the first round. Number of Voters (13) Rank5431 firstACCA secondCAAC C wins!

Wait…what do you mean C won?!?! A won the original election and the only change was favorable to A! There is a failure in Monotonicity with the Hare System

Plurality Runoff Method (procedure) A runoff method (new election using the same ballots), where the two candidates with the most first place votes are pitted head-to head. This method is not monotone. Number of Voters (13) Rank4432 firstABCD secondBADC thirdCCAA fourthDDBB A and B tie with four first place votes each

Plurality Runoff Number of Voters (13) Rank4432 firstABAA secondBABB A wins. Who would have won with the Hare system?? Number of Voters (13) Rank4432 firstABCC secondBAAA thirdCCBB In the first round only D would have been eliminated.

Practice Problems pg (skills check) also, exercises 5-10

With so many different ways to count votes, and with so many different winners, how do we tell who the true winner is???

Arrow’s Impossibility Theorem

Approval Voting